# Orbital Energy

1. Jan 30, 2005

### decamij

Just want to confirm. If an object is in orbit,, around earth, for example, the its total energy is equal to half of its Eg. But what is its kinetic energy equal to? Is there another expression rather than Ek = 1/2mv^2?

2. Jan 30, 2005

### da_willem

For circular motion:

$$E_k = \frac{1}{2}m\frac{(2 \pi r)^2}{T^2}$$

With T the period of the orbital motion. Now if you equate the gravitational force to the centripetal force $mv^2/r$ and use the same expression as above for the velocity you get:

$$T^2=\frac{(2 \pi r)^2 r}{GM}$$

Wich is Keplers law for circular orbits. Filling this expression in the kinetic energy expression:

$$E_k = \frac{GMm}{2r}=-\frac{E_g}{2}$$

This shouldn't be too surprising as total energy is kinetic + potential energy and you already noticed this equalled half the potential energy.
$$E_{pot}=E_g=-G\frac{Mm}{r}$$